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Cohen and Set Theory

Published online by Cambridge University Press:  15 January 2014

Akihiro Kanamori*
Affiliation:
Department of Mathematics, Boston University, 111 Cummington Street, Boston, Massachusetts 022215, USAE-mail: aki@math.bu.edu

Abstract

We discuss the work of Paul Cohen in set theory and its influence, especially the background, discovery, development of forcing.

Type
Research Article
Copyright
Copyright © Association for Symbolic Logic 2008

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References

REFERENCES

[1] Albers, Donald J., Alexanderson, Gerald L., and Reid, Constance (editors), More mathematical people, Harcourt Brace Jovanovich, Boston, 1990.Google Scholar
[2] Ax, James and Kochen, Simon, Diophantine problems over local fields II. A complete set of axioms for p-adic number theory, American Journal of Mathematics, (1965), pp. 631648.Google Scholar
[3] Banach, Stefan, Über die Barie'sche Kategorie gewisser Funktionenmengen, Studia Mathematica, vol. 3 (1931), p. 174.Google Scholar
[4] Bell, John L., Boolean-valued models and independence proofs in set theory, Oxford Logic Guides, Clarendon Press, Oxford, 1977, second ed., 1985; third ed., 2005.Google Scholar
[5] Blass, Andreas R. and Scedrov, Andre, Freyd's models for the independence of the axiom of choice, Memoirs of the American Mathematical Society, vol. 79 (1989), no. 404, p. 134 pp.Google Scholar
[6] Chen, Jingrun, On the presentation of a large even number as the sum of a prime and the product of at most two primes, Kexue Tongbao (Chinese Science Bulletin), vol. 17 (1966), pp. 385388, in Chinese.Google Scholar
[7] Cohen, Paul J., On Greens theorem, Proceedings of the American Mathematical Society, vol. 10 (1959), pp. 109112.Google Scholar
[8] Cohen, Paul J., On a conjecture of Littlewood and idempotent measures, American Journal of Mathematics, vol. 82 (1960), pp. 191212.Google Scholar
[9] Cohen, Paul J., A note on constructive methods in Banach algebras, Proceedings of the American Mathematical Society, vol. 12 (1961), pp. 159163.Google Scholar
[10] Cohen, Paul J., Idempotent measures and homomorphisms of group algebras, Proceedings of the International Congress of Mathematicians (Stockholm, 1962), Institut Mittag-Leffler, Djursholm, 1963, pp. 331336.Google Scholar
[11] Cohen, Paul J., The independence of the continuum hypothesis I, Proceedings of the National Academy of Sciences of the United States of America, vol. 50 (1963), pp. 11431148.Google Scholar
[12] Cohen, Paul J., A minimal model for set theory, Bulletin of the American Mathematical Society, vol. 69 (1963), pp. 537540.Google Scholar
[13] Cohen, Paul J., The independence of the continuum hypothesis II, Proceedings of the National Academy of Sciences of the United States of America, vol. 51 (1964), pp. 105110.Google Scholar
[14] Cohen, Paul J., Independence results in set theory, The theory of models, Proceedings of the 1963 international symposium at Berkeley (Addison, John W. Jr., Henkin, Leon, and Tarski, Alfred, editors), Studies in Logic and the Foundations of Mathematics, North-Holland, Amsterdam, 1965, pp. 3954.Google Scholar
[15] Cohen, Paul J., Set theory and the continuum hypothesis, W. A. Benjamin, New York, 1966.Google Scholar
[16] Cohen, Paul J., Decision procedures for real and p-adic fields, Communications on Pure and Applied Mathematics, (1969), pp. 131151.Google Scholar
[17] Cohen, Paul J., Comments on the foundations of set theory, Axiomatic set theory (Scott, Dana S., editor), Proceedings of Symposia in Pure Mathematics, vol. 13, American Mathematical Society, Providence, 1971, pp. 915.Google Scholar
[18] Cohen, Paul J., Automorphisms of set theory, Proceedings of the Tarski symposium, Proceedings of Symposia in Pure Mathematics, vol. 25, American Mathematical Society, Providence, 1974, pp. 325330.Google Scholar
[19] Cohen, Paul J., The discovery of forcing, Rocky Mountain Journal of Mathematics, vol. 32 (2002), no. 4, pp. 10711100.Google Scholar
[20] Cohen, Paul J., Skolem and pessimism about proof in mathematics, Philosophical Transactions of The Royal Society, vol. 363 (2005), pp. 24072418.Google Scholar
[21] Cohen, Paul J., My interaction with Kurt Gödel: The man and his work, Horizons of truth, Cambridge University Press, Cambridge, 2009, in honor of the 100th annniversay year of Kurt Gödel (2006).Google Scholar
[22] Dummett, Michael, Elements of intuitionism, Oxford Logic Guides, vol. 2, Clarendon Press, Oxford, 1977.Google Scholar
[23] Easton, William B., Proper class of generic sets, Notices of the American Mathematical Society, vol. 11 (1964), p. 205, abstract.Google Scholar
[24] Cohen, Paul J., Powers of regular cardinals, Annals of Mathematical Logic, vol. 1 (1970), pp. 139178, abridged version of Ph.D. thesis, Princeton University 1964.Google Scholar
[25] Feferman, Solomon, Some applications of the notions of forcing and generic sets, Fundamenta Mathematicae, vol. 56 (1965), pp. 325345.Google Scholar
[26] Cohen, Paul J., Some applications of the notions of forcing and generic sets (summary), The theory ofmodels, Proceedings of the 1963 international symposium at Berkeley (Addison, John W. Jr., Henkin, Leon, and Tarski, Alfred, editors), Studies in Logic and the Foundations of Mathematics, North-Holland, Amsterdam, 1965, pp. 8995.Google Scholar
[27] Feferman, Solomon and Levy, Azriel, Independence results in set theory by Cohen's method II, Notices of the American Mathematical Society, vol. 10 (1963), p. 593, abstract.Google Scholar
[28] Freyd, Peter J., The axiom of choice, Journal of Pure and Applied Algebra, vol. 19 (1980), pp. 103125.Google Scholar
[29] Gödel, Kurt F., Consistency prooffor the generalized continuum hypothesis, Proceedings of the National Academy of Sciences of the United States of America, vol. 25 (1939), pp. 220224.Google Scholar
[30] Gödel, Kurt F., The consistency of the axiom of choice and of the generalized continuum hypothesis with the axioms of set theory, Annals of Mathematics Studies, no. 3, Princeton University Press, Princeton, 1940.Google Scholar
[31] Gödel, Kurt F., Collected works, volume IV: Correspondence A-G, (Feferman, Solomon and Dawson, John W. Jr., editors), Clarendon Press, Oxford, 2003.Google Scholar
[32] Halpern, James D., The independence of the axiom of choice from the Boolean prime ideal theorem, Fundamenta Mathematicae, vol. 55 (1964), pp. 5766.CrossRefGoogle Scholar
[33] Halpern, James D. and Läuchli, Hans, A partition theorem, Transactions of the American Mathematical Society, vol. 124 (1966), pp. 360367.Google Scholar
[34] Halpern, James D. and Levy, Azriel, The ordering theorem does not imply the axiom of choice, Notices of the American Mathematical Society, vol. 11 (1964), p. 56, abstract.Google Scholar
[35] Gödel, Kurt F., The Boolean prime ideal theorem does not imply the axiom of choice, Axiomatic set theory (Scott, Dana S., editor), Proceedings of Symposia in Pure Mathematics, vol. 13, American Mathematical Society, Providence, 1971, pp. 83134.Google Scholar
[36] Kechris, Alexander S. and Louveau, Alain, Descriptive set theory and the structure of sets of uniqueness, London Mathematical Society Lecture Note Series, vol. 128, Cambridge University Press, Cambridge, 1987.Google Scholar
[37] Lawvere, F. William, Quantifiers and sheaves, Actes du Congres International des Mathématiciens (Nice 1970), vol. 1, Gauthier-Villars, Paris, 1971, pp. 329334.Google Scholar
[38] Lerman, Manuel, Degrees of unsolvability, Perspectives in Mathematical Logic, Springer-Verlag, Berlin, 1983.Google Scholar
[39] Levy, Azriel, Independence results in set theory by Cohens method I, Notices of the American Mathematical Society, vol. 10 (1963), pp. 592593, abstract.Google Scholar
[40] Levy, Azriel, Independence results in set theory by Cohens method III, Notices of the American Mathematical Society, vol. 10 (1963), p. 593, abstract.Google Scholar
[41] Levy, Azriel, Independence results in set theory by Cohens method IV, Notices of the American Mathematical Society, vol. 10 (1963), p. 593, abstract.Google Scholar
[42] Levy, Azriel, Definability in axiomatic set theory I, Logic, methodology and philosophy of science. Proceedings of the 1964 international congress at Jerusalem (Bar-Hillel, Yehoshua, editor), Studies in Logic and the Foundations of Mathematics, North-Holland, Amsterdam, 1965, pp. 127151.Google Scholar
[43] Levy, Azriel, Definability in set theory II, Mathematical logic and foundations of set theory. Proceedings ofan international colloquium (Bar-Hillel, Yehoshua, editor), Studies in Logic and the Foundations of Mathematics, North-Holland, Amsterdam, 1970, pp. 129145.Google Scholar
[44] Moore, Gregory H., The origins of forcing, Logic Colloquium '86 (Drake, Frank R. and Truss, John K., editors), Studies in Logic and the Foundations of Mathematics, North-Holland, Amsterdam, 1988, pp. 143173.Google Scholar
[45] Nasar, Sylvia, A beautiful mind, Simon and Schuster, New York, 1998.Google Scholar
[46] Oxtoby, John C., Measure and category, a survey of the analogies between topological and measure spaces, Graduate Texts in Mathematics, vol. 2, Springer-Verlag, New York, 1971.Google Scholar
[47] Scott, Dana S., Measurable cardinals and constructible sets, Bulletin de l'Académie Polonaise des Sciences, Seérie des Sciences Matheématiques, Astronomiques et Physiques, vol. 9 (1961), pp. 521524.Google Scholar
[48] Shepherdson, John C., Inner models of set theory—part III, The Journal of Symbolic Logic, vol. 18 (1953), pp. 145167.CrossRefGoogle Scholar
[49] Shoenfield, Joseph R., Unramified forcing, Axiomatic set theory (Scott, Dana S., editor), Proceedings of Symposia in Pure Mathematics, vol. 13, American Mathematical Society, Providence, 1971, pp. 357381.Google Scholar
[50] Skolem, Thoralf, Einige Bemerkungen zur axiomatischen Begründung der Mengenlehre, Matematikerkongressen i Helsingfors den 4-7 juli 1922, Den femte skandinaviska matematikerkongressen, Redogörelse, Akademiska-Bokhandeln, Helsinki, 1923, pp. 217232.Google Scholar
[51] Smale, Stephen, Differentiable dynamical systems, Bulletin of the American Mathematical Society, vol. 73 (1967), pp. 747817.Google Scholar
[52] Solovay, Robert M., 2No can be anything it ought to be, The theory of models, Proceedings of the 1963 international symposium at Berkeley (Addison, John W. Jr., Henkin, Leon, and Tarski, Alfred, editors), Studies in Logic and the Foundations of Mathematics, North-Holland, Amsterdam, 1965, abstract, p. 435.Google Scholar
[53] Solovay, Robert M., The measure problem, Notices of the American Mathematical Society, vol. 12 (1965), p. 217, abstract.Google Scholar
[54] Solovay, Robert M., Real-valued measurable cardinals, Notices of the American Mathematical Society, vol. 13 (1966), p. 721.Google Scholar
[55] Solovay, Robert M., A model of set theory in which every set of reals is Lebesgue measurable, Annals of Mathematics, vol. 92 (1970), pp. 156.Google Scholar
[56] Solovay, Robert M., Real-valued measurable cardinals, Axiomatic set theory (Scott, Dana S., editor), Proceedings of Symposia in Pure Mathematics, vol. 13, American Mathematical Society, Providence, 1971, pp. 397428.Google Scholar
[57] Solovay, Robert M. and Tennenbaum, Stanley, Iterated Cohen extensions and Souslin's problem, Annals of Mathematics, vol. 94 (1971), pp. 201245.Google Scholar
[58] Spector, Clifford, On degrees of recursive unsolvability, Annals of Mathematics, vol. 64 (1956), pp. 581592.Google Scholar
[59] Solovay, Robert M., Measure-theoretic construction of incomparable hyperdegrees, The Journal of Symbolic Logic, vol. 23 (1958), pp. 280288.Google Scholar
[60] Steel, John R., Forcing with tagged trees, Annals of Mathematical Logic, vol. 15 (1978), pp. 5574.Google Scholar
[61] Tennenbaum, Stanley, Souslin s problem, Proceedings of the National Academy of Sciences of the United States of America, vol. 59 (1968), pp. 6063.Google Scholar
[62] Tierney, Myles, Sheaf theory and the continuum hypothesis, Toposes, algebraic geometry and logic, Lecture Notes in Mathematics, vol. 274, Springer, Berlin, 1972, pp. 1342.Google Scholar
[63] Vopěnka, Petr, The general theory of ∇-models, Commentationes Mathematicae Universitatis Carolinae, vol. 8 (1967), pp. 145170.Google Scholar
[64] Yandell, Benjamin H., The honors class. Hilbert's problems and their solutions, A K Peters, Natick, 2002.Google Scholar